Homogeneous Kobayashi-hyperbolic manifolds with high-dimensional group of holomorphic automorphisms (Q2286180)

In this elegant paper, the author classifies all connected homogeneous Kobayashi-hyperbolic manifolds of complex dimension \(n\) greater than or equal to 2 whose automorphism group has (real) dimension \(n^2-2\). To understand the significance of the number \(n^2-2\), recall, firstly, that if \(M\) is a connected complex manifold of dimension \(n\), then \(\mathrm{Aut}(M)\) -- the group of holomorphic automorphisms of \(M\) -- is a real Lie group of dimension\,\(\leq n^2+2n\). Let \(d(M):= \dim\mathrm{Aut}(M)\). In a series of papers, the author has classified all connected Kobayashi-hyperbolic manifolds \(M\) of dimension \(n\geq 2\) with \(n^2-1\leq d(M)\leq n^2+2n\). As a part of this effort, he showed that \(d(M)\) cannot take the values \(n^2+3, n^2+4,\dots, n^2+2n-1\). The value \(n^2-2\) is critical in the sense that one cannot hope to explicitly list (up to biholomorphic equivalence) all \(M\) with \(d(M)=n^2-2\): for instance, a generic Reinhardt domain \(M\varsubsetneq \mathbb{C}^2\) has \(d(M) = 2\). However, as the paper under review shows, the classification becomes tractable when \(d(M) = n^2-2\) and \(M\) is homogeneous. The classification is very elegant: with \(M\) as stated above, \(M\) is biholomorphic either to \(B^2\times B^1\times B^1\) (in which case \(n=4\)) or to \(B^3\times B^2\) (in which case \(n=5\)). Here, \(B^d\) denotes the open Euclidean unit ball in \(\mathbb{C}^d\). The proof of the above theorem relies on the main result of \textit{È. B. Vinberg} and \textit{I. I. Piatetski-Shapiro} [Trans. Mosc. Math. Soc. 12, 404--437 (1963); translation from Tr. Mosk. Mat. O.-va 12, 359--388 (1963; ; Zbl 0137.05603)] and its generalization to (Kobayashi hyperbolic) manifolds in [\textit{K. Nakajima}, J. Math. Kyoto Univ. 25, 269--291 (1985; Zbl 0583.32066)]. The author also revisits his classification when \(d(M)\) satisfies \(n^2-1\leq d(M)\leq n^2+2n\) to provide an explicit list of all connected homogeneous Kobayashi-hyperbolic manifolds \(M\) of complex dimension \(n\) at least 2 for which \(n^2-2\leq d(M)\leq n^2+2n\).